The well known Wiener-Kuratowski explicit definition of the ordered pair, which sets ⟨x, y⟩ = {{x}, {x, y}}, works well in many set theories but fails for those with classes which cannot be members of singletons. With the aid of the Axiom of Foundation, we propose a recursive definition of ordered pair which addresses this shortcoming and also naturally generalizes to ordered tuples of greater lenght. There are many advantages to the new definition, for it allows for uniform definitions working (...) equally well in a wide range of models for set theories. In ZFC and closely related theories, the of an ordered pair of two infinite sets under the new definition turns out to be equal to the maximum of the ranks of the sets. (shrink)

, Rudolf Carnap became a chief proponent of the doctrine that the statements of intuitionism carry nonstandard intuitionistic meanings. This doctrine is linked to Carnap's ‘Principle of Tolerance’ and claims he made on behalf of his notion of pure syntax. From premises independent of intuitionism, we argue that the doctrine, the Principle, and the attendant claims are mistaken, especially Carnap's repeated insistence that, in defining languages, logicians are free of commitment to mathematical statements intuitionists would reject. I am grateful to (...)Nathan Carter, Gary Ebbs, Janet Folina, Luise Prior McCarty, Stewart Shapiro, Neil Tennant, Christopher Tillman, Beth Tropman, Wen-fang Wang, and two anonymous referees for their comments and suggestions. CiteULike Connotea Del.icio.us What's this? (shrink)

Straightforwardly and strictly intuitionistic inferences show that the BrouwerKolmogorov (BHK) interpretation, in the presence of a formulation of the recognition principle, entails the validity of the Law of Testability: that the form s original weak counterexample reasoning was fallacious. The results of the present article extend and refine those of McCarty, C. (2012). Antirealism and Constructivism: Brouwer’s Weak Counterexamples. The Review of Symbolic Logic. First View. Cambridge University Press.

[David Charles] Aristotle, it appears, sometimes identifies well-being (eudaimonia) with one activity (intellectual contemplation), sometimes with several, including ethical virtue. I argue that this appearance is misleading. In the Nicomachean Ethics, intellectual contemplation is the central case of human well-being, but is not identical with it. Ethically virtuous activity is included in human well-being because it is an analogue of intellectual contemplation. This structure allows Aristotle to hold that while ethically virtuous activity is valuable in its own right, the best (...) life available for humans is centred around, but not wholly constituted by, intellectual contemplation. /// [Dominic Scott] In Nicomachean Ethics X 7-8, Aristotle distinguishes two kinds of eudaimonia, primary and secondary. The first corresponds to contemplation, the second to activity in accordance with moral virtue and practical reason. My task in this paper is to elucidate this distinction. Like Charles, I interpret it as one between paradigm and derivative cases; unlike him, I explain it in terms of similarity, not analogy. Furthermore, once the underlying nature of the distinction is understood, we can reconcile the claim that paradigm eudaimonia consists just in contemplation with a passage in the first book requiring eudaimonia to involve all intrinsic goods. (shrink)

: A standard interpretation of Kantian "maxims" sees them as expressing reasons for action, implying that we cannot act without a maxim. But recent challenges to this interpretation claim that Kant viewed acting on maxims as optional. Kant's understanding of maxims derives from Christian Wolff, who regarded maxims as major premises of the practical syllogism. This supports the standard interpretation. Yet Kant also viewed commitments to maxims as essential for virtue and character development, which supports challenges to the standard interpretation, (...) and raises questions about the coherence of Kant's overall conception of the role of maxims in practical philosophy. (shrink)

The present paper is one installment in a lengthy task, the replacement of atomistic interpretations of Wittgenstein's Tractatus by a wholistic interpretation on which the world-in-logical-space is not constructed out of objects but objects are abstracted from out of that space. Here, general arguments against atomism are directed toward a specific target, the four aspects of the atomistic reading of Tractatus given in the Hintikkas' Investigating Wittgenstein (Hintikka & Hintikka 1986). The aspects in question are called the semantical, metaphysical, epistemological (...) and formal.What follows a précis of the Hintikkas' rendering of Wittgenstein's perspective is a characterization of the wholistic interpretation, comparing Wittgenstein's world and the transcendental conditions it sets upon possible notation to a blank page and the conditions it sets upon what is about to be written there. There will not be occasion to bring arguments against each plank in the atomist's platform or in support of each facet of wholism. But there is an extended treatment of the first two aspects — the semantical and metaphysical — which takes off from Wittgenstein's determination that, in his hands, logic must take care of itself. (shrink)

The comparative seriousness of business law and business ethics gives some business people the impression that there is nothing important in business ethics. The costly penalties of illegal conduct compared to the uncertain consequences of unethical conduct support a common illusion that business ethics is much less important than law for business people. To dispel the illusion I distinguish two perspectives from which we can view the relation of business and normative systems: the internal and external perspectives. I show that (...) in one perspective, ethics is hardly less important than law, and in the other perspective it is more important, more fundamental than law. I conclude with a discussion of why business persons must place ethical and legal rules ahead of profits. (shrink)

Let S be a deductive system such that S-derivability (⊦s) is arithmetic and sound with respect to structures of class K. From simple conditions on K and ⊦s, it follows constructively that the K-completeness of ⊦s implies MP(S), a form of Markov's Principle. If ⊦s is undecidable then MP(S) is independent of first-order Heyting arithmetic. Also, if ⊦s is undecidable and the S proof relation is decidable, then MP(S) is independent of second-order Heyting arithmetic, HAS. Lastly, when ⊦s is many-one (...) complete, MP(S) implies the usual Markov's Principle MP. An immediate corollary is that the Tarski, Beth and Kripke weak completeness theorems for the negative fragment of intuitionistic predicate logic are unobtainable in HAS. Second, each of these: weak completeness for classical predicate logic, weak completeness for the negative fragment of intuitionistic predicate logic and strong completeness for sentential logic implies MP. Beth and Kripke completeness for intuitionistic predicate or sentential logic also entail MP. These results give extensions of the theorem of Gödel and Kreisel (in [4]) that completeness for pure intuitionistic predicate logic requires MP. The assumptions of Godel and Kreisel's original proof included the Axiom of Dependent Choice and Herbrand's Theorem, no use of which is explicit in the present article. (shrink)

The project of antirealism is to construct an assertibility semantics on which (1) the truth of statements obeys a recognition condition so that (2) counterexamples are forthcoming to the law of the excluded third and (3) intuitionistic formal predicate logic is provably sound and complete with respect to the associated notion of validity. Using principles of intuitionistic mathematics and employing only intuitionistically correct inferences, we show that prima facie reasonable formulations of (1), (2), and (3) are inconsistent. Therefore, it should (...) not be assumed that the project of anti-realism as it bears upon intuitionistic mathematics and logic can be accomplished. (shrink)

This article is an exercise in computational jurisprudence. It seems clear thatthe field of AI and Law should draw upon the insights of legal philosophers,whenever possible. But can the computational perspective offer anything inreturn? I will explore this question by focusing on the concept of OWNERSHIP,which has been debated in the jurisprudential literature for centuries. Althoughthe intellectual currents here flow mostly in one direction – from legal philosophy to AI – I will show that there are also some insights to (...) be gained from a computational analysis of the OWNERSHIP relation. In particular, the article suggests a computational explanation for the emergence of abstract property rights, divorced from concrete material objects. (shrink)

This significant, stimulating contribution to Kantian practical philosophy strives to interpret Kant’s theory of action in ways that will increase readers’ understanding and appreciation of Kant’s moral theory. Its thesis is that Kant combines metaphysical freedom and psychological determinism: our actions within the phenomenal world are causally determined by our prior psychological states in that world and are appearances of our free action in the noumenal world. McCarty argues for a metaphysical, “two-worlds” interpretation of Kant’s transcendental distinction between appearances (...) and things in themselves over epistemological or methodological “two-standpoints” interpretations familiar from Christine Korsgaard. (shrink)